| L(s) = 1 | + (−1.36 − 0.381i)2-s + (0.472 + 0.864i)3-s + (1.70 + 1.03i)4-s + (1.07 − 1.96i)5-s + (−0.312 − 1.35i)6-s + (0.0329 + 0.0440i)7-s + (−1.92 − 2.06i)8-s + (1.09 − 1.70i)9-s + (−2.21 + 2.25i)10-s + (2.79 − 1.27i)11-s + (−0.0924 + 1.96i)12-s + (−4.11 − 3.08i)13-s + (−0.0280 − 0.0725i)14-s + (2.20 + 0.00351i)15-s + (1.83 + 3.55i)16-s + (−1.83 + 0.131i)17-s + ⋯ |
| L(s) = 1 | + (−0.962 − 0.269i)2-s + (0.272 + 0.499i)3-s + (0.854 + 0.519i)4-s + (0.480 − 0.876i)5-s + (−0.127 − 0.554i)6-s + (0.0124 + 0.0166i)7-s + (−0.682 − 0.731i)8-s + (0.365 − 0.568i)9-s + (−0.699 + 0.714i)10-s + (0.843 − 0.385i)11-s + (−0.0266 + 0.568i)12-s + (−1.14 − 0.855i)13-s + (−0.00750 − 0.0193i)14-s + (0.568 + 0.000906i)15-s + (0.459 + 0.888i)16-s + (−0.444 + 0.0318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.898024 - 0.561342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898024 - 0.561342i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.381i)T \) |
| 5 | \( 1 + (-1.07 + 1.96i)T \) |
| 23 | \( 1 + (-3.82 - 2.89i)T \) |
| good | 3 | \( 1 + (-0.472 - 0.864i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-0.0329 - 0.0440i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-2.79 + 1.27i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (4.11 + 3.08i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (1.83 - 0.131i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (3.72 + 4.30i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.435 - 0.377i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.88 - 6.42i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.681 + 3.13i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-4.72 + 3.03i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (0.741 + 1.35i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-1.27 + 1.27i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.63 - 7.52i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (0.156 + 1.08i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.57 + 1.93i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.00 - 2.69i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-6.29 - 2.87i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (14.4 + 1.03i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-1.27 - 8.88i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (1.19 + 5.48i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (0.866 - 2.95i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.43 - 0.964i)T + (88.2 + 40.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.57776680219110758969442059809, −9.886879219713590019604617640442, −8.905555523880776546017610228053, −8.819967640591150309839606228780, −7.35308186921247041853843409355, −6.44711052173188134292562198243, −5.09272739650916002444117067429, −3.85611652211466007042941966078, −2.50976037718187989252623311851, −0.899688628808894641870500854255,
1.77571728813135060844070255439, 2.56149251419845820137307270346, 4.49580282552702539634722527579, 6.08148922461175199319571106487, 6.87030120472882148143985484168, 7.41446322697073140501745723266, 8.472003410968977183199983831228, 9.519431658689836442961607186273, 10.11592363049598379453162889608, 11.01246587073131754187137084534
